# transformations replaced by symmetry

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8 matching pages ♦

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## 8 matching pages

##### 1: 19.15 Advantages of Symmetry

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►Symmetry in $x,y,z$ of ${R}_{F}(x,y,z)$, ${R}_{G}(x,y,z)$, and ${R}_{J}(x,y,z,p)$
replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17).
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##### 2: 19.22 Quadratic Transformations

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##### 3: 19.25 Relations to Other Functions

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19.25.16
$$\mathrm{\Pi}(\varphi ,{\alpha}^{2},k)=-\frac{1}{3}{\omega}^{2}{R}_{J}(c-1,c-{k}^{2},c,c-{\omega}^{2})+\sqrt{\frac{(c-1)(c-{k}^{2})}{({\alpha}^{2}-1)(1-{\omega}^{2})}}{R}_{C}(c({\alpha}^{2}-1)(1-{\omega}^{2}),({\alpha}^{2}-c)(c-{\omega}^{2})),$$
${\omega}^{2}={k}^{2}/{\alpha}^{2}$.

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##### 4: Bille C. Carlson

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►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few.
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##### 5: Errata

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Chapter 18 Orthogonal Polynomials
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Figure 4.3.1
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Section 27.20
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Subsection 21.10(i)
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Equations (18.16.12), (18.16.13)
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This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.

*Reported 2015-11-12 by James W. Pitman.*

The entire Section was replaced.

The entire original content of this subsection has been replaced by a reference.

The upper and lower bounds given have been replaced with stronger bounds.

##### 6: 18.7 Interrelations and Limit Relations

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###### §18.7(i) Linear Transformations

… ►###### §18.7(ii) Quadratic Transformations

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18.7.17
$${U}_{2n}\left(x\right)={W}_{n}\left(2{x}^{2}-1\right),$$

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18.7.18
$${T}_{2n+1}\left(x\right)=x{V}_{n}\left(2{x}^{2}-1\right).$$

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##### 7: 18.17 Integrals

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►and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1.
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###### §18.17(v) Fourier Transforms

… ►###### Jacobi

… ►###### §18.17(vi) Laplace Transforms

… ►###### §18.17(vii) Mellin Transforms

…##### 8: 20.11 Generalizations and Analogs

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►If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):
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